Nuprl Lemma : WtoSet-order-preserving
∀[A:Type]. ∀[B:A ⟶ Type].
∀x,y:W(A;a.B[a]). ∀b:𝔹. ((x Wcmp(A;a.B[a];b) y) ⇒ (WtoSet(a.B[a];x) Wcmp(Type;a.a;b) WtoSet(a.B[a];y)))
Proof
Definitions occuring in Statement :
WtoSet: WtoSet(a.B[a];x),
Wcmp: Wcmp(A;a.B[a];leq),
W: W(A;a.B[a]),
bool: 𝔹,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
infix_ap: x f y,
subtype_rel: A ⊆r B,
Set: Set{i:l},
all: ∀x:A. B[x],
WtoSet: WtoSet(a.B[a];x),
Wsup: Wsup(a;b),
Wcmp: Wcmp(A;a.B[a];leq),
mk-set: f"(T),
bool: 𝔹,
ifthenelse: if b then t else f fi ,
exists: ∃x:A. B[x]
Lemmas referenced :
W-induction,
all_wf,
W_wf,
bool_wf,
Wcmp_wf,
WtoSet_wf,
subtype_rel_self,
Wsup_wf,
btrue_wf,
mk-set_wf,
bfalse_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
functionEquality,
universeEquality,
because_Cache,
independent_functionElimination,
lambdaFormation,
unionElimination,
dependent_functionElimination,
productElimination,
dependent_pairFormation
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type].
\mforall{}x,y:W(A;a.B[a]). \mforall{}b:\mBbbB{}.
((x Wcmp(A;a.B[a];b) y) {}\mRightarrow{} (WtoSet(a.B[a];x) Wcmp(Type;a.a;b) WtoSet(a.B[a];y)))
Date html generated:
2018_05_22-PM-09_52_07
Last ObjectModification:
2018_05_16-PM-01_31_46
Theory : constructive!set!theory
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